Filtered spectral differentiation method for numerical differentiation of periodic functions with application to heat flux estimation

Bazán, Fermín S. V.; Bedin, Luciano

doi:10.1007/s40314-019-0968-4

Cited by 5 publications

(2 citation statements)

References 29 publications

(39 reference statements)

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“…21 деформівного твердого тіла. Ефективність використання аналітико-числових методів показана при розв'язуванні багатьох задач сучасної математичної фізики: [2] -використовують інтерполяційні функції Лобатто -Лежандра в задачі геодезичної гравіметрії; [3] -використовують усічені ряди Фур'є в розв'язку задачі поширення теплового потоку в спіральних трубах; [4] -при вивченні руху електронів в атомах та іонах у сферичній системі координат в якості базису були використані гамільтонові власні функції і сферичні гармоніки, тобто пов'язані з ними функції Лежандра; [5] -при розв'язуванні рівняння адвекціїдисперсії та дифузії з дробовою похідною за часом використовували дуальний базис Бернштейна.…”

Section: проектирование летательных аппаратовunclassified

Числове Дослідження Базисних Систем При Розв’язуванні Аналітико-Числовими Методами Крайових Задач На Відрізку

Халілов¹,

Ткаченко²,

Бондарева³

et al. 2019

aktt

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Spectral methods have indisputable advantages over numerical methods in solving various problems of mathematical physics. The advantages are the high convergence and accuracy of approximate solutions, which is most relevant for calculating the strength of aerospace machinery. The problem of choosing basic functions inevitably arises when applying spectral methods. The problem is that, in addition to providing exponential convergence, the system of basic functions has to satisfy some other requirements: stability of approximate solutions and procedures for their obtaining, reduction of calculations, convenience, and some more. This paper compares eleven basic systems: the systems constructed in the form of linear combinations of Legendre polynomials that satisfy either only the main boundary conditions or the main and natural ones, similar systems constructed using Chebyshev polynomials, and the functions proposed by Khalilov S. A., systems of Lagrange – Lobatto interpolation polynomials using the Legendre and Chebyshev interpolation points, system of trigonometric functions, exponentiation, and system of finite functions of the finite element method. The convergence speed of the approximate solution to the exact one, the error in the equations of boundary value problems, and the condition numbers of matrices of linear algebraic equation systems, which arise when using variational, projection and collocation methods, were compared. The study performed on three test problems modeling beam bending: classic beam bending under unevenly distributed load, bending of additionally stretched beam on the elastic basis and geometrically nonlinear bending. The impact of the Gibbs effect on the approximate solution convergence is investigated. Among the considered basic systems, the system of basis functions in the form of linear combinations of Legendre polynomials has proved to be the best, as they satisfy all boundary conditions. This basis leads to the highest speed at which approximate solution approaches the exact one, the error in the equations approaches to zero, and also it has the smallest increase in the condition number with the increase in the order of SLE matrices, which appear due to variational and projection methods. The finite functions of the finite element method have proved to be the worst in terms of accuracy and convergence.

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Section: проектирование летательных аппаратовunclassified

Числове Дослідження Базисних Систем При Розв’язуванні Аналітико-Числовими Методами Крайових Задач На Відрізку

Халілов¹,

Ткаченко²,

Бондарева³

et al. 2019

aktt

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“…Numerical differentiation [13][14][15][16] approximates the function value of the unknown objective function at certain points based on the information of the known finite discrete sampling points. According to the literature, numerical differentiation mainly includes: finite difference type [17], polynomial interpolation type [13,15], regularization method [18,19], and undetermined coefficients [20].…”

Section: Introductionmentioning

confidence: 99%

Research and Application of PID Controller with Feedforward Filtering Function

Wang¹,

Shen²

2022

Intelligent Electronics and Circuits - Terahertz, ITS, and Beyond

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Most of the existing differential methods focus on the differential effect and do not make full use of the differential link’s filtering effect of reducing order and smoothing. In Proportion Integral Differential (PID) control, the introduction of differential can improve the dynamic performance of the system. However, the actual differential (containing differential gain) will be subject to the impact of high-frequency noises. Therefore, this paper proposes a differential with filtering function, which has weak effect on noise amplification, and strong effect on reducing order and smoothing. Firstly, a discrete differentiator was constructed based on the Newton interpolation, and the concept of “algorithm bandwidth” was defined to ensure the differential effect. Then, the proposed algorithm was used to design a new PID controller with feedforward filtering function. In the experiments, the proposed PID controller is applied to a high-performance hot water supply system. The result shows that the system obtains better control quality. It verifies that the proposed PID controller has a feedforward filtering function and can effectively remove high-frequency noise.

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